2017 USAMO Problems

Day 1

Note: For any geometry problem whose statement begins with an asterisk ( $*$ ), the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement will result in an automatic 1-point deduction.

Problem 1

Prove that there are infinitely many distinct pairs $(a,b)$ of relatively prime positive integers $a > 1$ and $b > 1$ such that $a^b + b^a$ is divisible by $a + b.$

Solution

Problem 2

Let $m_1, m_2, \ldots, m_n$ be a collection of $n$ positive integers, not necessarily distinct. For any sequence of integers $A = (a_1, \ldots, a_n)$ and any permutation $w = w_1, \ldots, w_n$ of $m_1, \ldots, m_n$ , define an $A$ -inversion of $w$ to be a pair of entries $w_i, w_j$ with $i < j$ for which one of the following conditions holds: $a_i \ge w_i > w_j$ $w_j > a_i \ge w_i$ , or $w_i > w_j > a_i$ . Show that, for any two sequences of integers $A = (a_1, \ldots, a_n)$ and $B = (b_1, \ldots, b_n)$ , and for any positive integer $k$ , the number of permutations of $m_1, \ldots, m_n$ having exactly $k$ $A$ -inversions is equal to the number of permutations of $m_1, \ldots, m_n$ having exactly $k$ $B$ -inversions.

Solution

Problem 3

( $*$ ) Let $ABC$ be a scalene triangle with circumcircle $\Omega$ and incenter $I$ . Ray $AI$ meets $\overline{BC}$ at $D$ and meets $\Omega$ again at $M$ ; the circle with diameter $\overline{DM}$ cuts $\Omega$ again at $K$ . Lines $MK$ and $BC$ meet at $S$ , and $N$ is the midpoint of $\overline{IS}$ . The circumcircles of $\triangle KID$ and $\triangle MAN$ intersect at points $L_1$ and $L_2$ . Prove that $\Omega$ passes through the midpoint of either $\overline{IL_1}$ or $\overline{IL_2}$ .

Solution

Day 2

Note: For any geometry problem whose statement begins with an asterisk ( $*$ ), the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement will result in an automatic 1-point deduction.

Problem 4

Let $P_1$ , $P_2$ , $\dots$ , $P_{2n}$ be $2n$ distinct points on the unit circle $x^2+y^2=1$ , other than $(1,0)$ . Each point is colored either red or blue, with exactly $n$ red points and $n$ blue points. Let $R_1$ , $R_2$ , $\dots$ , $R_n$ be any ordering of the red points. Let $B_1$ be the nearest blue point to $R_1$ traveling counterclockwise around the circle starting from $R_1$ . Then let $B_2$ be the nearest of the remaining blue points to $R_2$ travelling counterclockwise around the circle from $R_2$ , and so on, until we have labeled all of the blue points $B_1, \dots, B_n$ . Show that the number of counterclockwise arcs of the form $R_i \to B_i$ that contain the point $(1,0)$ is independent of the way we chose the ordering $R_1, \dots, R_n$ of the red points.

Solution

Problem 5

Let $\mathbf{Z}$ denote the set of all integers. Find all real numbers $c > 0$ such that there exists a labeling of the lattice points $( x, y ) \in \mathbf{Z}^2$ with positive integers for which: only finitely many distinct labels occur, and for each label $i$ , the distance between any two points labeled $i$ is at least $c^i$ .

Solution

Problem 6

Find the minimum possible value of $\frac{a}{b^3+4}+\frac{b}{c^3+4}+\frac{c}{d^3+4}+\frac{d}{a^3+4}$ given that $a$ , $b$ , $c$ , $d$ are nonnegative real numbers such that $a+b+c+d=4$ .

Solution

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

2017 USAMO (Problems • Resources)
Preceded by 2016 USAMO	Followed by 2018 USAMO
1 • 2 • 3 • 4 • 5 • 6
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2017 USAMO Problems

Contents

Day 1

Problem 1

Problem 2

Problem 3

Day 2

Problem 4

Problem 5

Problem 6